Recall that every finitely generated module over a regular local ring is Tor-rigid [1]}, [2]}. Therefore part (i) follows from Proposition REF  by setting \(\operatorname{\mathcal {Z}}=\mathfrak {m})\) . Moreover, parts (ii) and (iii) are immediate consequence of Theorem REF , and part (iv) is a special case of part (iii).
